# How do you show that (3+sqrt2)/(5+sqrt8) can be written to (11-sqrt2)/17?

Aug 12, 2018

#### Explanation:

We know that ,

color(red)((1)a^2-b^2=(a-b)(a+b)

Let ,

$X = \frac{3 + \sqrt{2}}{5 + \sqrt{8}}$

Multiplying numerator and denominator by $\left(5 - \sqrt{8}\right)$

$X = \frac{3 + \sqrt{2}}{5 + \sqrt{8}} \times \frac{5 - \sqrt{8}}{5 - \sqrt{8}}$

Simplifying we get

$X = \frac{3 \left(5 - \sqrt{8}\right) + \sqrt{2} \left(5 - \sqrt{8}\right)}{{\left(5\right)}^{2} - {\left(\sqrt{8}\right)}^{2}} \to \textcolor{red}{\left[\because \left(1\right)\right]}$

$\therefore X = \frac{15 - 3 \sqrt{8} + 5 \sqrt{2} - \sqrt{16}}{25 - 8}$

$\therefore X$=(15-3*2sqrt2+5sqrt2- 4)/17to[becausesqrt8=sqrt(4xx2)=2sqrt2]

$\therefore X = \frac{11 - 6 \sqrt{2} + 5 \sqrt{2}}{17}$

$\therefore X = \frac{11 - \sqrt{2}}{17}$

Aug 13, 2018

By Rationalization

#### Explanation:

$\frac{3 + \sqrt{2}}{5 + \sqrt{8}}$

By Rationalization, Note: $\frac{a}{x + \sqrt{y}} = \frac{a}{x + \sqrt{y}} \times \frac{x - \sqrt{y}}{x - \sqrt{y}}$ or vice versa..

Hence;

$\frac{3 + \sqrt{2}}{5 + \sqrt{8}} \times \frac{5 - \sqrt{8}}{5 - \sqrt{8}}$

Rationalizing..

$\frac{\left(3 + \sqrt{2}\right) \left(5 - \sqrt{8}\right)}{\left(5 + \sqrt{8}\right) \left(5 - \sqrt{8}\right)}$

Expanding..

$\frac{\left(15 - 3 \sqrt{8} + 5 \sqrt{2} - \sqrt{16}\right)}{\left(25 - 8\right)}$

Simplifying..

$\frac{\left(15 - 3 \sqrt{8} + 5 \sqrt{2} - 4\right)}{17}$

Further simplifying..

$\frac{\left(15 - 3 \sqrt{8} + 5 \sqrt{2} - 4\right)}{17}$

Collecting like terms..

$\frac{\left(15 - 4 - 3 \sqrt{8} + 5 \sqrt{2}\right)}{17}$

Simplifying..

$\frac{\left(11 - 3 \sqrt{2 \times 4} + 5 \sqrt{2}\right)}{17}$

Further simplifying..

$\frac{\left(11 - 3 \left(\sqrt{2} \times \sqrt{4}\right) + 5 \sqrt{2}\right)}{17}$

$\frac{\left(11 - 3 \left(\sqrt{2} \times 2\right) + 5 \sqrt{2}\right)}{17}$

$\frac{\left(11 - 3 \times 2 \left(\sqrt{2}\right) + 5 \sqrt{2}\right)}{17}$

$\frac{\left(11 - 6 \sqrt{2} + 5 \sqrt{2}\right)}{17}$

$\frac{\left(11 - \sqrt{2}\right)}{17}$

Therefore;

$\frac{3 + \sqrt{2}}{5 + \sqrt{8}} = \frac{11 - \sqrt{2}}{17}$

As required..